XIX International Conference on Electrical Machines - ICEM 2010, Rome
Iterative FEM-BEM Technique for an Efficient
Computation of Magnetic Fields in Regions
with Ferromagnetic Bodies
I. R. Ciric, M. Maricaru, I. F. Hantila, and S. Marinescu
Abstract - Inside the ferromagnetic bodies the magnetic
field is determined by applying the Finite Element Method
(FEM) with a Dirichlet boundary condition for the magnetic
potential. The resulting tangential component of the field
intensity is used as boundary condition for the exterior field
problem whose solution is obtained by the Boundary Element
Method (BEM) which provides the new boundary condition for
the interior problem. The nonlinearity of the highly permeable
ferromagnetic material is treated by implementing the
Polarization Fixed Point Method (PFPM), where the magnetic
polarization is corrected in terms of the field intensity. While
preserving the intrinsic advantages of the FEM and the BEM,
separately, the proposed technique also allows to easily take
into consideration the terminal voltage of the coils as input data
and a simple field solution for multiply connected structures,
such as transformers, electrical machines, and a multitude of
electromagnetic devices.
Index Terms-boundary integral equations, hybrid
FEM-BEM techniques, iterative methods, multiply connected
regions, nonlinear media.
I. INTRODUCTION
D
IFFICULTIES encountered when employing the FEM
for the analysis of fields in unbounded regions or in
structures with moving bodies, as well as those related
to parasitic sources artificially generated at the interfaces
between the discretization elements, with undesirable
implications in force calculations, have determined the
development of various integral and hybrid methods [1 ]-[7].
The application of the PFPM [8] has allowed the extension
of integral formulations to nonlinear media with periodic
eddy currents and moving bodies [9]-[1 O].
In this work, a novel iterative FEM-BEM procedure is
proposed for an efficient solution of magnetic field problems
in unbounded regions in the presence of ferromagnetic
bodies. A vector potential formulation is illustrated for
two-dimensional fields in simply and multiply connected
structures. The field problems inside and outside the
ferromagnetic regions are solved separately and repeatedly
with the interior field determined by applying the FEM
under Dirichlet boundary conditions and the exterior one by
This work was supported in part by the Natural Sciences and
Engineering Research Council of Canada and the Romanian National
Council of Scientific Research under Grant ID_2010.
I. R. Ciric is with the Department of Electrical and Computer
Engineering, The University of Manitoba, Winnipeg, MB, R3T 5V6 Canada
(e-mail: irciric@ee.urnanitoba.ca).
M. Maricaru and I. F. Hantila are with the Department of Electrical
Engineering, Politehnica University of Bucharest, Spl. Independentei 313,
Bucharest, 060042 Romania (e-mail: hantila@elth.pub.ro).
S. Marinescu is with the Research Institute for Electrical Engineering, Spl.
Unirii 313. Bucharest 030138 Romania le-mail: stelian.marinescu!alicoe.ro).
the BEM under Neumann boundary conditions, the solution
in each region providing successively the boundary
condition for the problem in the other region. A highly rate
of convergence is insured by treating the material
nonlinearity with the PFPM, where the magnetic
polarization is corrected in terms of the field intensity.
II. FIELD PROBLEM IN LINEAR MAGNETIC BODIES
Assuming a null current density, the magnetic vector
potential A'= kA' for a two-dimensional structure satisfies
the following equation inside the region n occupied by
linearized ferromagnetics:
(1)
where k is the unit vector in the longitudinal direction, µ is
the magnetic permeability, and I represents a given
distribution of magnetic polarization. The potential A' is
known on the boundary
of 0. We represent A' in n
an
as
A'=
I
A'; 'P; +
ie{n}
I
(2)
A'; 'P;
ie{ns}
where {n} and {n 8 } represent the sets of nodes of the
discretization mesh inside n and on an ' respectively' and
'I'; and A'; are the linear nodal element function and the
potential values at the node i. Applying the FEM yields
Ia 1;A'; = - Ia 1;A'; +t J,
ie{n}
j
E
{n}
(3)
ie{ns}
with
(4)
(5)
The values A'; are known for i e {n 8 } and the matrix of
a Ji 's in (3) is sparse, symmetric and positive definite. The
memory space for the matrix entries is practically
proportional to n and the solution of (3) is performed very
rapidly using various sparse matrix techniques. From the
solution of (3) one first obtains A'; at the interior nodes and
then the tangential field intensity on
an .
III. FIELD PROBLEM IN FREE SPACE
First, assume the ferromagnetic region to be simply
connected. The potential integral equation on the boundary
an of the free space unbounded region no is obtained
from
aA(r)=
f
(R·n)A(r')dl'-§ ln_!_oA(r')dl'+A 0 (r)
2
an R an
an R
(6)
Fig. I. Standard notation adopted.
where a is the solid angle under which a small
neighborhood of n 0 is seen from the observation point, r
where M' and N' are 1x n 8 matrices.
and r' are the position vectors of the observation and the
= r - r', R = n is the unit
source points, respectively,
R
IRI,
vector in the direction normal to the boundary (see Fig. 1),
and A 0 is the vector potential due to the given current
distribution,
IV.
ITERATIVE PROCEDURE
At the boundary
an between no and n
we connect the
vector potential and its normal derivative in the two regions
by imposing the continuity condition for the vector potential
and for the tangential component of field intensity, i.e.
Ao(r)=.uo~[Jln ~dS'-f In ~dS']=B(Qi.(r)-Q_(r))
S+
{11)
S_
(7)
for a total B of Ampere-turns of the coils, considered to be
uniformly distributed over the cross sections of areas S +
and S _ of the coil sides carrying current in the direction of
k and -k, respectively, with S+ = S_
=S
(see Fig. 2). For
coupling the BEM applied to (6) with the FEM in Section II,
the vector potential is taken to have a linear variation on
each boundary element with a constant normal derivative.
Locating the observation point at the boundary nodes, (6) is
transformed into
aA
M=MA+N-+BC
an
where 11 is the tangential component of the polarization.
The bigger the value of relative permeability µr, the
stronger a contraction will result from ( 11) for the normal
derivatives of the potential. Thus, the following iterative
process is proposed:
a)
having known aA/ an on the elements of
the nodes of
an
an,
A'= A at
are obtained from (9), i.e. by BEM;
b)
from A' on the boundary, one determines A' in
the FEM and, then, oA'/on on an;
c)
oA/an in a) is corrected with (I 1).
n
by
This cycle is repeated until the difference between aA/ an
(8)
for two successive iterations satisfies an error condition, for
instance
where the symbols A, aA/an, and Care now used to denote
the matrices nB x I of the boundary values of A and
oA/ on
in (6) and of Q+ -Q_ in (7), A is the diagonal matrix of
the values of a at the boundary nodes, and M and N are
some n 8 xn 8 matrices. Equation (8) gives A when oA/on
is known,
aA
A=Z-+BT
an
with
Z
= (A-M)-I N
and T
observation point inside
no
=(A-M)- 1C.
(9)
{12)
with £ conveniently small. The matrices Z and T are
computed only once, before the start of the iterative process.
When the terminal voltages of the coils are given, instead
of the coil currents, the magnetic fluxes of the coils are
known. For illustrating the computation procedure in this
case, consider a system with only one coil. The average
magnetic flux per coil tum is obtained from (10) as
At any
the vector potential is
(13)
calculated from
2tt4(r) = M'(r)A + N'(r) aA +B(Q+(r)-Q_ (r)) (10)
an
with the entries of the 1x n 8
obtained from
matrices M <l> and N <l>
5
J
1
J
(14)
Mil> = - ( M'(r)dS- M'(r)dSJ.
27TS s
s+
1
NIJ> = 27TS
(Js
N'(r)dS-
+
J
s
ng
s._
I
I
I
I
(15)
N'(r)dSJ
5
135
-
I
I
I
I
pi
s·+
~r
:n
:n
on;
and
!.¥
M
O>
....
n~
n
one
145
(16)
48
241
Taking into account (9), we get
Fig. 2. Multiply connected region for a single-phase transformer.
-
oA
Cl>=W-+BY
(17)
on
therefore, fix a zero value of reference for the potential at a
on;
where W = M <l>Z + N <1> and Y = M <l>T + Q<l> . We remark
chosen node pi on
that if two different values of B' say B' and B"' correspond
other n: = nk -1 nodes from
(according to (17)) to
ii>•
and
ii>",
respectively, then the
differences t:i..8 = B'-8" and t:i..<i> = <1> 1-<i>"
related as
are simply
(20)
(18)
As a consequence, the step a) in the above iterative
procedure is modified as follows. For an arbitrary B0 , <1> 0
is determined with (17) and, then, to the imposed value of
ii>
and determine the potentials at the
corresponds a 8 =Bo + t:i..B, with t:i..B determined from
(18), i.e. 118 = t:i..<i>/Y, t:i..<i> = <i>-<1> 0 . This value of 8 is
where
zi
respectively,
with A;1 ,
are n! x nk
and Ti
Mf
zi
and
= (A;1 -M{
N{
and
f1 N{,
T;
n! x I matrices,
= (A;1 -M{
f1
Cl+•
being the matrices in (8) but for only
n;
nodes, and C 1+ is the
values Q+ in (7).
n: x I
matrix of the boundary
used in (9) and we continue with step b). W is a Ix n B
In the case when the coil current is given, for known
matrix and Ya scalar which are, again, computed only once,
before starting the iterative process.
values of the normal derivative, the potential on an.e is
V.
TREATMENT OF MULTIPLY CONNECTED REGION
To explain the technique proposed for the field analysis
in multiply connected regions, consider the magnetic circuit
of a single-phase transformer, sketched with only one of its
coils in Fig. 2. On the boundaries
an.e
and
on;
of the two
nonmagnetic regions O.~ and O.~ we consider, respectively,
n8 and nk nodes. As in (9), for O.~ we have
obtained with (19) and that on on; with (20), the latter up
to an unknown additive constant c0 . The potential in 0.
satisfies the equation (see (3))
LaJiA';=- Iaj;A'; -co Ia 1; +t 1 ,
ie{n)
ie{ns)
je{n} (21)
ie{nkJ
where n B = ns + n~. A' in 0. can be determined as
follows. First, without any additive constant, we apply the
FEM in 0. and obtain the potential A" from (3), with the
boundary conditions given by A"= Ae on an.e and A"= A;
(19)
but only with the coil side of cross-sectional area
s_. ze
is
on on;, Ae and A; being those computed from (19) and
(20), respectively.
In general, this solution corresponds to an incorrect total
8" of Ampere-turns, i.e. 8"-:;: 8 ,
a n8 x ns matrix and Te a n8 x l matrix. For the inner
subregion n~' the potential is determined up to an additive
constant and the matrix A; - M; is now singular. Let's,
_I
µo
f . 8A"
di= 8"
an
an'
(22)
Then, we obtain the potential Ac in
n
corresponding to a
ane '
a unity value of it on
zero potential on
null polarization, from
oni
6.5
12
12
6.5
/
and to a
O>
j
,La1;Ac=- ,Laji,
ie{n}
E
{n}
(23)
ie{n1}
O>
i.e. (21) with c0 = 1 and t J = 0. This Ac corresponds to a
n
(Jc Ampere-turns of the coil. The correct potential A' in
is given by
A'= A"+Ac f!i.{J
(Jc
(24)
O>
where f!i.(J = (J - ()". Ac is calculated only once before
starting iterating.
For the case when the coil terminal voltage is given, we
notice that, while the potential at the point pi on
anj
O>
/
is
·...•
\...
\
equal to zero for the problem in n~' the actual potential at
_
.
.... .
',
'\
__
/
,1
'-..
\
Fig. 3. Electromagnet (dimensions are in mm).
pi has now a value c 0 ::t: 0 to be determined. The average
magnetic flux per tum is evaluated by applying (17) to the
interior and exterior subregions, and Ampere's theorem to
the boundaries
-
anj
and
j
aAi
ane '
i
ct> =W - - Y -
an
f='
1.8
.---.---.--,---,--,---,.---,-~---.---~
1.6
l---+--.L_-J.i----!,....=J--+-*"=~=1:=:::t==1t=1
1.4 t--1_-+.y;"---+----11-----+--+--+--+--l---4----+-----'
ar 1.2 l-z-+-+----11---+--+--t---+--+--+---i--+--
1 .( aAi
µo
J' -dl+c0
on'0
an
.§
1.0 t-7t--+----1r----r---t--t---+--+--+---l--+-~
ti
0.8
::I
~
0.6
0.4
(25)
1/---t--t---t--t--+-+--t--+--+--+------<
1----t--t---t--t--+-+---l--+--+--+------<
1----t--t---t--t--+-+---l--+--+--+------<
0.2 t----t---t--+--t--+--+---1---1----+--+---J
0.0
_
_ __,__....___._ _.__....1...._.1.-......1
~__..
0
_.__~__..
2
3
4
5
6
7
8
9
10
11
Field intensity H(kA/m)
This yields the constant c 0 which has to be added to the
potential in n~ computed from (20).
VI.
Fig. 4. B-H relationship.
11/(m) - 1<m-lf = J_!_(J<m) -/(m-1) JdS < t:J ll1(m)ll2
µ
FIELD ANALYSIS IN NONLINEAR MEDIA
n.µ
µ
The field problem in nonlinear ferromagnetic regions is
solved iteratively by employing the PFPM [8]. The
nonlinear constitutive relation B=F(H) is replaced by
The fictitious permeability µ is chosen such that G is a
(26)
contraction. For instance, in the case of isotropic media,
and I= f(H)- µH = g(H), it is
where I= I Hf H
B=µH+l
where µ is a constant and the additive term /, playing the
(28)
necessary that µ >
.!.(!!i..)
.This condition insures the
2 dH
max
role of a magnetic polarization, is corrected iteratively in
terms of H from
convergence of the iterative procedure for the solution of the
linear field problem presented in the previous Sections.
I= F(H)-µH = G(H)
VII. ILLUSTRATIVE EXAMPLES
(27)
Starting with some distribution of/, a magnetic field (B, H)
is determined using (26) and, then, I is corrected using (27).
These two steps are repeated until the norm of the difference
between the polarizations in two successive iterations is
sufficiently small
The magnetic field of the electromagnet in Fig. 3 is
produced by a coil with 5,000 turns carrying a direct current
of 49 mA. The thickness of the ferromagnetic region is
19.5 mm and the stacking factor of the laminations is 0.92.
For the B-H characteristic shown in Fig. 4, the minimum and
the maximum values of the differential oermeabilitv arr.
g:
3.0
2.7
2.4
2.1
1.8
values extend now into the saturation zone, a smaller value
of µ in (26) was chosen, µ = (4µmax +4µmin)/9
~
~
~ 1.5
0
LL
> µmax/2, even though the PFPM convergence is insured
)'..
_S
1.2
"1'..
0.9
0.6
...,...__
-
0.3
0.0
1.5
2.0
2.5
3.0
3.5
4.5
4.0
5.0
Airgap (mm)
Fig. 5. Attractive force for the electromagnet in Fig. 3.
for any value µ > µmax/2. The shape of the magnetization
current is presented in Fig. 6. The FEM mesh used for this
example had 1309 nodes and 2370 triangular elements, and
the time required to perform the field analysis for 48 time
steps was 216 s employing the same notebook as in the first
example. Due to the magnetic saturation, the number of
iterations in the PFPM procedure climbed in some cases to
87, while for the hybrid FEM-BEM procedure three
iterations sufficed, in most of the cases even one iteration
being sufficient, for the same accuracy as in the first
example.
VIII. CONCLUSIONS
0.8 ,.--.,----,-----,----,-----,Vl----:::i:;=
........
c-rb..---,-,----,----,,-----,----,
0.7 l--~+--+---+-/-E---1--+-,,,,...~+-b..--+--+--t---l---l
~
o.6 1-----1---l-+--+v-+--+----l--+_i-+--+-+---+----l
~ 0.5
7
\
~ 0.4 l-----l---l--+17-l-+--l--t---+--l--+~\--+--l--l---j
8
0.3
l---j--+-_i_-l--+--+--l-----l--+--1-4\+--+--l-----l
0.2
1-----j--+v-+-+-+---+--+---+--+----f---*~--+--+----i
0.1.~
~
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Phase (rad)
Fig. 6. Half of a period of the magnetization current for the transfonner in
Fig. 2.
µmin = 12µo and µmax = 9947 µ 0 , respectively, and, thus,
µ in (26) was chosen µ = (4µmax + µmin)/5 > µmax/2. A
sketch of the magnetic induction field lines is drawn in Fig.
3 for an airgap of 5 mm and the dependence on the airgap of
the attraction force between the armatures is presented in
Fig. 5. For an airgap of 3.4 mm, the value of the computed
force is 0.61 N, while the measured value is 0.62 N,
differences between computed results and actual values
being partially due to the two-dimensional model adopted. A
FEM mesh with 1691 nodes and 3009 triangular elements
was used. The computation for I 8 positions of the mobile
armature required a time of 247 s when using a I .66 GHz
(Intel Atom) processor notebook. For each position, the
PFPM iterations continued until an error less then E: 1 =I o-4
was achieved, with only about six iterations being necessary.
The iterative FEM-BEM procedure required at most four
iterations for an error less then E: = I 0-4 (see Section IV and
(12)).
A second example is that of the transformer in Fig. 2,
where the region occupied by the ferromagnetic material is
doubly connected. The transformer is operated on no-load
with a sinusoidal with time voltage of maximum value of
329.5 V at the terminals of a 325 tum winding. This gives
directly the magnetic flux of the winding when the voltage
drop corresponding to its resistance is negligible. The
ferromagnetic core has a thickness of 48 mm and a stacking
factor of the laminations of 0.9, with the same B-H
characteristic as in Fig. 4. Since the magnetic induction
The main
contribution to the
computational
electro111agnetics of the work presented consists in the
efficient iterative procedure for the solution of the equations
in the hybrid FEM-BEM for structures containing nonlinear
ferromagnetic bodies. The rapid convergence of the
procedure is determined by the big difference between the
permeabilities of the ferromagnetic bodies and that of the
air. This difference is preserved by treating the nonlinearity
of B-H relationship with the PFPM, whose convergence is
insured when the correction of the magnetic polarization is
made in terms of the magnetic field intensity and the
constant µ in (26) is chosen to be greater than half of the
maximum value of· the differential permeability in the
operating zone. The field problem inside the ferromagnetic
bodies is solved by implementing the FEM under Dirichlet
boundary conditions, the system matrix being sparse,
symmetric and positive definite. The required memory space
and the computation time are very small. The boundary
conditions are corrected by solving the exterior field
problem applying the BEM where the stiffiless matrix is
dense, but is calculated only once, before the start of the
iterations.
As well, an extension of the proposed procedure to field
problems for multiply connected regions is presented, where
the BEM applied to bounded nonmagnetic regions allows
the determination of the vector potential up to an additive
constant. It should be remarked that the technique developed
in this paper deals easily with situations when either the coil
currents or their terminal voltages are given, thus the
flux-current relationships necessary for the modeling of field
structures by lumped parameter circuits being efficiently
established.
In the proposed procedure, all the advantages of the FEM
are kept regarding the system matrix structure and, at the
same time, all the advantages of the BEM are exploited. It is
also advantageous to apply the procedure to structures with
moving bodies, since only the entries of the BEM matrices
corresponding to the coupling between the bodies in relative
motion are to be recalculated for various specific positions.
In the air regions the magnetic field equations are exactly
satisfied, thus avoiding the fictitious generation of the
parasitic current sheets specific to the FEM. As a
consequence, when calculating forces, the flux of the
magnetic stress tensor is rigorously independent of the
integration surfaces.
IX.
[I]
REFERENCES
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X.
BIOGRAPHIES
loan R. Ciric is a Professor of Electrical Engineering at the University of
Manitoba, Winnipeg, MB, Canada. His major research interests are in the
mathematical modeling of stationary and quasi-stationary fields,
electromagnetic fields in the presence of moving solid conductors and
levitation, field theory of special electrical machines, de corona ionized
fields, analytical and numerical methods for wave scattering and diffraction
problems, propagation along waveguides with discontinuities, transients,
and inverse problems. He published numerous scientific papers and a few
electrical engineering textbooks, and has been a contributor to book
chapters on magnetic field modeling and electromagnetic wave scattering by
complex bodies.
Dr. Ciric is a Fellow of the IEEE and a member of the Electromagnetics
Academy. He is listed in Who's Who in Electromagnetics.
Mihai Maricaru graduated
Bucharest and obtained his
university in 2007, where
research interests are mainly
hybrid methods).
in 2001 from the Politehnica University of
Ph.D. in Electrical Engineering at the same
he is currently an Assistant Professor. His
in electromagnetic field computation (integral,
Florea I. Hantila received the Electrical Engineer degree in 1967 and the
Ph.D. degree in 1976, both from the Politehnica University of Bucharest,
where he is currently a Professor and the head of the Department of
Electrical Engineering. His teaching and research interests are in nonlinear
electromagnetic field analysis and in numerical methods.
Stelian Marinescu graduated in 1971 from the Politehnica University of
Bucharest. His activity started immediately after as a scientific researcher at
Research Institute for Electrical Engineering, where he is currently working.
His fields of expertise include research, design and construction of special
electrical machines.